Analytical model for relativistic corrections to the nuclear magnetic shielding constant in atoms

Physics Letters A 353 (2006) 190–193 www.elsevier.com/locate/pla

Analytical model for relativistic corrections to the nuclear magnetic shielding constant in atoms Rodolfo H. Romero ∗ , Sergio S. Gomez Facultad de Ciencias Exactas, Universidad Nacional del Nordeste, Avenida Libertad 5500 (3400), Corrientes, Argentina Received 21 November 2005; received in revised form 19 December 2005; accepted 22 December 2005 Available online 29 December 2005 Communicated by B. Fricke

Abstract We present a simple analytical model for calculating and rationalizing the main relativistic corrections to the nuclear magnetic shielding constant in atoms. It provides good estimates for those corrections and their trends, in reasonable agreement with accurate four-component calculations and perturbation methods. The origin of the effects in deep core atomic orbitals is manifestly shown. © 2005 Elsevier B.V. All rights reserved. PACS: 31.30.Jv; 32.30.Dx; 33.25.+k Keywords: Nuclear magnetic shielding; Relativistic effects; NMR parameters

It is nowadays well established that the theoretical calculation of the spectral parameters of nuclear magnetic resonance (NMR) spectroscopy requires the kinematic description of the theory of relativity for properly accounting them when the system contains heavy atoms [1–3]. This is so because, roughly speaking, the speed of the inner shell electrons is a sizable fraction of that of light. In particular, the relativistic calculation of the nuclear magnetic shielding σ , in molecules containing heavy atoms, has recently been addressed by a number of formalisms and methods at different levels of approximation [4–15]. They established that the relativistic corrections are, to a large extent, independent of the molecular environment. It has also been shown that this behavior arises from the fact that those corrections are due, mainly, to the electrons belonging to core orbitals. From those calculations, several systematic observations were made, such as a power-law dependence of the relativistic corrections on the nuclear charge ∼ Z 3 [6,9, 10], the predominant contribution of the s atomic orbitals, and the decreasing relative contribution of the external orbitals as compared to the inner one, 1s [9,10]. Although those results

* Corresponding author.

E-mail address: [email protected] (R.H. Romero). 0375-9601/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.12.081

look sound and plausible, they have not received a theoretical foundation. It is the purpose of this Letter to show that those properties can be understood and accurately estimated by using a simple analytical model. The interaction energy of the magnetic dipole moment μ = γ h¯ I of a nuclear spin I in an external magnetic field B is  μi (δij − σij )Bj , E=− (1) ij

where γ is the magnetogyric ratio of the isotope considered, δij is the Kronecker delta and σij is the nuclear magnetic shielding tensor. In atoms, σij is diagonal and a shielding constant σ , equal to one third of the trace of the nuclear magnetic shielding tensor, can be defined, i.e., σij =

∂ 2E , ∂μi ∂Bj

σ=

1 Tr σij . 3

(2)

This parameter describes the effect of the induced field produced by the electronic motion modifying the external magnetic field at the nucleus position. The theory for diamagnetic fields in atoms was given by Lamb [16] and the theory for the magnetic shielding of nuclei in molecules was originally discussed by Ramsey in one of his

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191

Table 1 Nuclear magnetic shielding constant of the rare gas atoms calculated with Eq. (7) compared to ab initio RPA calculations from Ref. [12]. All values are given in ppm  d 2 Z Atom Configuration σind Ref. [12] n (1/n ) 2 10 18 36 54 86

He Ne Ar Kr Xe Rn

1s 2 1s 2 2s 2 2p 6 [Ne] 3s 2 3p 6 [Ar] 3d 10 4s 2 4p 6 [Kr] 4d 10 5s 2 5p 6 [Xe] 4f 14 5d 10 6s 2 6p 6

2.00 4.00 4.89 6.50 7.44 8.94

seminal papers [17]. In this work, we propose an independentparticle model based on the exact solutions of the hydrogenlike atom for the calculation of the relativistic corrections to the nuclear magnetic shielding in atoms. We use atomic units (m = h¯ = e = 1) throughout this Letter; in these units, the fine structure constant α = 1/c  137.036; when convenient, the explicit dependence on the electron mass and charge of the derived expressions will be shown. We shall derive first, from a slightly different point of view, Lamb’s expression for the diamagnetic contribution to the shielding constant in atoms. The Hamiltonian for a hydrogenlike atom in the presence of both an external (AB ) and the nuclear magnetic field (AN ) is 1 Ze2 (p − eA)2 − 2m r Ze2 e2 A2 e p2 − + − A · p, = (3) 2m r 2m m where A = AB + AN . The first two terms in Eq. (3) form the Schrödinger Hamiltonian of the hydrogen-like atom, H0 , having energy eigenvalues En = −Z 2 /2n2 . The third A2 -term contains bilinear products of the form (1), thus contributing to the (diamagnetic) nuclear magnetic shielding tensor through firstorder perturbation theory. Last term in Eq. (3) is both linear in μ and B, contributing to the energy through second-order perturbation theory; this (paramagnetic) term vanishes in atoms because of the central symmetry. Hence, disregarding that term, the Hamiltonian (3) for an atom can also be written, up to firstorder terms, as

71.0 710.0 1562.3 4153.5 7131.2 13646.9

– 550.30 1237.84 3245.70 5642.43 10728.20

μ · Bα 2 /3m. Hence, the exact energy levels will be given by En∗ = −Z ∗2 /2n2 or, up to first-order terms, Z2 Zα 2 + μ · B, (6) 2n2 3mn2 and the eigenfunctions, correct through first-order in the Zeeman-like diamagnetic interaction σ μ · B, are hydrogen-like d wave functions ψnm depending on μ and B through the effec∗ tive charge Z . Eq. (6) shows that an electron in the level n of a hydrogenic atom contributes σn = 17.75Z/n2 parts per million (ppm) to its nuclear magnetic shielding. Then, an independentparticle model of the atom predicts that the shielding constant σ will be given by En∗ = En + σn μ · B = −

occ  1 ppm, n2 n

H=

σindep = 17.75Z

H = H0 + H d = H0 + σ μ · B,

where the summation runs over all occupied states. Table 1 shows the values of σindep for the rare gas atoms compared to quantum-chemical calculations within the random phase approximation (RPA) of the polarization propagator taken from Ref. [12]. σindep systematically overestimates σRPA ; this fact can be attributed to the effect of the electronic correlation arising from the electron–electron interaction. In a many-electron atom, the inner electronic shells shield the external ones, thus giving a smaller effective nuclear charge for the outer electrons. As an estimation of the effect of the repulsive interaction between two electrons in the 1s level of a He-like atom on σ , one can account for it by shifting the nuclear charge Z → Z − 5/16 [18]. Then, for He-like atoms, σindep → σrep = 2(Z − 5/16) × 17.75 ppm, i.e.,

(4)

Hd

(7)

where is the perturbation Hamiltonian giving rise to the diamagnetic shielding, and

σrep = (35.5Z − 11.1) ppm,

e2 α 2 1 (5) 3m r is an isotropic operator to be used only in first-order perturbation theory. Lamb [16] estimated the expectation value of that operator by using the Thomas–Fermi model of the atom. We shall proceed in a different way to show a procedure that will prove to be useful in treating the relativistic corrections to the diamagnetic shielding constant. Because of the 1/r-dependence of σ , one can formally recast the Hamiltonian (4) into a hydrogen-like atomic Hamiltonian, by absorbing the term H d into the Coulomb potential for a rescaled effective nuclear charge Z ∗ = Z −

where the subscript rep refers to the repulsive interaction included. Then, the inclusion of the Coulomb repulsion reduces σindep by a constant value of 11.1 ppm in a two-electron atom independently of its nuclear charge. Eq. (8) gives 627.9, 1266.9, 1905.9, and 3041.9 ppm for Z = 18, 36, 54 and 86, which compare remarkably well to RPA calculations giving 627.92, 1266.93, 1905.95 and 3042.00 ppm. However, one should be cautious about generalizing this method for N -electron atoms (N > 2); for many electron atoms, one should include the repulsion between all different combinations of shells as well as the exchange interaction which is not present here; even if possible, that would complicate the simple model to such an extent that no simple picture could be extracted from it, which is its

σ=

(8)

192

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Table 2 Contribution from ns orbitals to the relativistic correction σ FC/SZ–KE , calculated with Eq. (15) and compared to ab initio results from Ref. [12]. All values are given in ppm Z

Atom

1s

2s

3s

4s

5s

6s

Total

Ref. [12]

10 18 36 54 86

Ne Ar Kr Xe Rn

11.3 65.9 527.2 1779.3 7187.4

1.4 8.2 65.9 222.4 898.4

–

– –

– – – 14.2 57.4

– – – – 33.3

12.7 76.5 620.8 2109.6 8555.0

12.43 72.39 596.62 2043.60 8254.63

2.4 19.5 65.9 266.2

main aim. The main usefulness of the model will be for properties weakly dependent on the electronic correlation, such as the relativistic corrections to σ [12,14]. Furthermore, a fully relativistic extension of Eq. (6) is straightforward. In a four-component relativistic framework, the diamagnetic shielding can be approximated as the expectation value of the operator σ [Eq. (5)] calculated with the ground state relativistic wave function [4,19]. Recasting again the Coulomb potential in the Dirac Hamiltonian by replacing Z → Z ∗ , expanding the energy eigenvalues up to O(Z ∗4 α 4 ) and, finally expanding Z ∗ up to first order in μ · B, one has d,rel the relativistic correction to the diamagnetic energy Enj = d,rel σnj μ · B, where   2Z 3 α 4 3 1 d,rel = − σnj (9) j + 1/2 4n 3n3 is the relativistic diamagnetic nuclear magnetic shielding of a hydrogen-like atom having its electron in the energy level (n, j ). The relativistic theory of the nuclear magnetic shielding constant predicts a nonvanishing paramagnetic contribution in atoms, in contrast to its nonrelativistic counterpart which predicts a vanishing paramagnetic contribution because of the central symmetry of the potential. To a large extent, that relativistic correction to the paramagnetic shielding can be accounted for through the so-called mass-velocity external-field term σ FC/SZ–KE [5]. This term comes from using secondorder perturbation theory with the perturbation operators spinZeeman kinetic energy H SZ–KE = −(α 2 /2)p 2 B · s and the Fermi contact hyperfine Hamiltonian H FC = (8π/3)α 2 δ(r)μ · s. We shall consider here an equivalent procedure to calculate σ FC/SZ–KE . The Hamiltonian for the hydrogen atom including the H SZ–KE interaction

p2 Ze2 α 2 2 (10) − − p B·s 2m r 2 can be formally recasted into a Hamiltonian for a hydrogen atom with a spin-dependent effective mass, Hs = p 2 /2m∗ − Ze2 /r, with   m  m 1 + α 2 mB · s , m∗ = (11) 2 1 − α mB · s

H = H0 + H SZ–KE =

where the fact that α 2  1 has been used in the last equation to approximate m∗ up to first order in the external field B. Then, we can evaluate the expectation value of the hyperSZ of H which fine Hamiltonian H FC with wave functions ψnlm s are identical to the eigenfunctions of the hydrogen atom except that m → m∗ , thus changing the Bohr’s radio from a0 to

8.2 27.8 112.3

a ∗ = h¯ 2 /m∗ e2 in the radial functions. Then, SZ 2  SZ FC SZ 8π 2 H ψ (0) , α μ · s ψnm E FC/SZ–KE = ψnm nm = 3

(12)

where SZ 2 ψ (0) =

3 Z3 Z 3 m3  1 + α 2 mB · s = πn3 a ∗3 πn3  Z 3 m3  2 1 + 3α  mB · s , (13) πn3 and only the s states ( = 0) contribute to the energy shift because of the delta function in H FC . The derivatives of the energy E FC/SZ–KE with respect to components of μ and B, Eq. (2), give the components of the tensor whose isotropic part is the correction σ FC/SZ–KE to the shielding constant, i.e., n00

σ FC/SZ–KE =

Z3 1  ∂ 2 E FC/SZ–KE = 2α 4 3 3 ∂μi ∂Bi n

(14)

i

for an electron in the level n of a hydrogenic atom, or 4α 4 Z 3 /n3 = 11.3 × 10−3 Z 3 /n3 ppm for the closed shell ns. Then, for our model of N -electron atom, the relativistic correction to the paramagnetic nuclear magnetic shielding will be given by occ occ   1 1 −3 3 = 11.3 × 10 Z ppm. 3 n n3 n n (15) FC/SZ – KE Table 2 shows the values of σ calculated for the noble

σ FC/SZ–KE = 4Z 3 α 2

gas atoms computed with Eq. (15) compared to the results of ab initio calculations from Ref. [12]. On the other hand, the leading-order scalar perturbative relativistic corrections to the diamagnetic nonrelativistic nuclear magnetic shielding are given by the derivatives of Eq. (2) with E being the energy corrections E d/Dw and E d/mv calculated within second-order perturbation theory with H d from Eq. (4) and the spin-free Darwin and mass velocity perturbation Hamiltonians α2π Zδ(r), 2 α2 H mv = − p 4 . 8

H Dw =

(16) (17)

We shall calculate E d/Dw and E d/mv , as precedently, by considering E d/Dw and E d/mv as the expectation values of the d of Hamiltonians (16) and (17) with the eigenfunctions ψnm the Hamiltonian H from Eq. (4), having the effective nuclear

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193

Table 3 Scalar Darwin and mass velocity relativistic corrections to the nuclear magnetic shielding constant of the rare gas atoms calculated with Eqs. (20) and (25) as compared to ab initio RPA calculations taken from Ref. [12]. All values are given in ppm Z

Atom

10 18 36 54 86

Ne Ar Kr Xe Rn

σ d/Dw

σ d/mv

Eq. (20)

Ref. [12]

Eq. (25)

Ref. [12]

−3.19 −19.25 −156.09 −530.38 −2149.61

−2.87 −17.39 −145.59 −502.24 −2059.89

5.92 36.96 309.54 1070.81 4428.75

4.95 30.84 267.55 943.06 3917.28

charge Z ∗ , namely,  d Dw d

2 π d 2 H ψ Z ψn00 , E d/Dw = ψnm nm = α 2 with   d 2 Z ∗3 1 Z2α2 3 ψ =  μ·B . Z − n00 m πn3 πn3

(18)

(19)

Hence, each electron contributes −Z 3 α 4 /2n3 to σ d/Dw , and the sum of contributions for doubly occupied ns shells in a manyelectron atom is given by σ

d/Dw

occ occ   1 1 −3 3 = −Z α = −2.84 × 10 Z ppm. 3 3 n n n n 3 4

Similarly, E d/mv

d |H mv |ψ d , where H mv = ψnm nm

(20)

H mv = −

2

(H − V )2 ,

(21)

V being the effective Coulomb interaction V (r) = −Z ∗ e2 /r. d = E ∗ ψ d , and the wellTaking into account that H ψnlm n nlm known identities for the H-atom  d 1 d Z∗ ψnm ψnm = 2 , (22) r n  d 1 d

Z ∗2 , ψnm 2 ψnm = 3 (23) r n ( + 1/2) we have d/mv

σn

=

  α4Z3 4 1 + − n 3( + 1/2) 2n3

d , and for an electron in the state ψnlm  occ  Z3α4  4 1 d/mv σ = − 4+ 3 2 n 3n ( + 1/2)

Acknowledgements

can be writ-

ten as α2

on the occupied orbitals and, henceforth, on the number of electrons Z. The relativistic corrections to the diamagnetic nuclear magnetic shielding (Table 3) have a definite sign, negative for σ d/Dw and positive for σ d/mv , thus partially canceling each other, as already noted in previous works [12,14]. It can be seen from Table 3 that σ d/mv is roughly twice |σ d/Dw | both for our model as for the results from ab initio calculations. That can be understood considering the leading terms from the two elecd/Dw d/mv = −Z 2 α 4 and σ1s = 2c1s Z 2 α 4 ; trons in the 1s shell σ1s they are in the relation 2c1s = 5/3. As already pointed above, the origin of the systematic overestimation of the absolute value of every magnitude within the model can be traced to the absence of electronic correlation in the model.

(24)

(25)

nm

for the entire atom. Table 3 shows the values of σ d/Dw and σ d/mv calculated with Eqs. (20) and (25) and compared to the results from ab initio calculations taken from Ref. [12]. The model presented in this work, allows to rationalize much of the systematics studied in previous works such as the Z 3 dependence of the 1s contribution to σ FC/SZ–KE reported in Ref. [10]. Indeed, the total value of every relativistic correction studied follows a power-law dependence ∼ Z 3 . The deviations from the strict cubic dependence on Z come from the summation on the quantum numbers (n, ) for each shell that depends

We acknowledge Prof. G.A. Aucar for valuable discussions and the critical reading of the manuscript. This work was partly supported by SeCyT, Universidad Nacional del Nordeste, Argentina. References [1] J. Mason (Ed.), Multinuclear NMR, Plenum, New York, 1987. [2] P. Pyykkö, Adv. Quantum Chem. 11 (1978) 353; P. Pyykkö, Chem. Rev. 88 (1988) 563. [3] J. Vaara, P. Manninen, P. Lantto, in: M. Kaupp, M. Bühl, V.G. Malkin (Eds.), Calculation of NMR and EPR Parameters: Theory and Applications, Wiley, New York, 2004, Chapter 13. [4] M.M. Sternheim, Phys. Rev. 128 (1962) 676. [5] H. Fukui, T. Baba, H. Inomata, J. Chem. Phys. 105 (1996) 3175. [6] H. Fukui, T. Baba, J. Chem. Phys. 108 (1998) 3854. [7] H. Quiney, H. Skaane, I.P. Grant, Chem. Phys. Lett. 290 (1998) 473. [8] L. Visscher, T. Enevoldsen, T. Saue, H.J.Aa. Jensen, J. Oddershede, J. Comput. Chem. 20 (1999) 1262. [9] S.S. Gomez, R.H. Romero, G.A. Aucar, J. Chem. Phys. 117 (2002) 7942. [10] S.S. Gomez, R.H. Romero, G.A. Aucar, Chem. Phys. Lett. 367 (2003) 265. [11] J. Vaara, P. Pyykkö, J. Chem. Phys. 108 (2003) 2973. [12] P. Manninen, P. Lantto, J. Vaara, K. Ruud, J. Chem. Phys. 119 (2003) 2623. [13] W. Kutzelnigg, Phys. Rev. A 67 (2003) 032109. [14] S.S. Gomez, J.I. Melo, R.H. Romero, G.A. Aucar, M.C. Ruiz de Azúa, J. Chem. Phys. 122 (2005) 064103. [15] L. Visscher, Adv. Quantum Chem. 48 (2005) 369. [16] W.E. Lamb Jr., Phys. Rev. 60 (1941) 817. [17] N.F. Ramsey, Phys. Rev. 78 (1950) 699. [18] L.D. Landau, E.M. Lifshitz, Quantum Mechanics, vol. 1, Pergamon, New York, 1965. [19] G.A. Aucar, T. Saue, L. Visscher, H.J.Aa. Jensen, J. Chem. Phys. 110 (1999) 6208.